Leray spectral sequence
In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. The formulation was of a spectral sequence, expressing the relationship holding in sheaf cohomology between two topological spaces X and Y, and set up by a continuous mapping
- f:X → Y.
At the time of Leray's work, neither of the two concepts involved (spectral sequence, sheaf cohomology) had reached anything like a definitive state. Therefore it is rarely the case that Leray's result is quoted in its original form. After much work, in the seminar of Henri Cartan in particular, a statement was reached of this kind: assuming some hypotheses on X and Y, and a sheaf F on X, there is a direct image sheaf
There are also higher direct images
The E2 term of the typical Leray spectral sequence is
- Hp(Y, Rqf∗F).
The required statement is that this abuts to the sheaf cohomology
- Hr(X, F).